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Properties

Label 1-66e2-4356.7-r0-0-0
Degree $1$
Conductor $4356$
Sign $0.0478 - 0.998i$
Analytic cond. $20.2291$
Root an. cond. $20.2291$
Motivic weight $0$
Arithmetic yes
Rational no
Primitive yes
Self-dual no
Analytic rank $0$

Origins

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Normalization:  

Dirichlet series

L(s)  = 1  + (0.964 + 0.263i)5-s + (−0.761 − 0.647i)7-s + (0.0665 − 0.997i)13-s + (0.736 + 0.676i)17-s + (0.198 − 0.980i)19-s + (−0.235 + 0.971i)23-s + (0.861 + 0.508i)25-s + (−0.00951 − 0.999i)29-s + (−0.345 + 0.938i)31-s + (−0.564 − 0.825i)35-s + (−0.466 − 0.884i)37-s + (0.290 − 0.956i)41-s + (0.0475 − 0.998i)43-s + (−0.820 + 0.572i)47-s + (0.161 + 0.986i)49-s + ⋯
L(s)  = 1  + (0.964 + 0.263i)5-s + (−0.761 − 0.647i)7-s + (0.0665 − 0.997i)13-s + (0.736 + 0.676i)17-s + (0.198 − 0.980i)19-s + (−0.235 + 0.971i)23-s + (0.861 + 0.508i)25-s + (−0.00951 − 0.999i)29-s + (−0.345 + 0.938i)31-s + (−0.564 − 0.825i)35-s + (−0.466 − 0.884i)37-s + (0.290 − 0.956i)41-s + (0.0475 − 0.998i)43-s + (−0.820 + 0.572i)47-s + (0.161 + 0.986i)49-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 4356 ^{s/2} \, \Gamma_{\R}(s) \, L(s)\cr =\mathstrut & (0.0478 - 0.998i)\, \overline{\Lambda}(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 4356 ^{s/2} \, \Gamma_{\R}(s) \, L(s)\cr =\mathstrut & (0.0478 - 0.998i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(1\)
Conductor: \(4356\)    =    \(2^{2} \cdot 3^{2} \cdot 11^{2}\)
Sign: $0.0478 - 0.998i$
Analytic conductor: \(20.2291\)
Root analytic conductor: \(20.2291\)
Motivic weight: \(0\)
Rational: no
Arithmetic: yes
Character: $\chi_{4356} (7, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((1,\ 4356,\ (0:\ ),\ 0.0478 - 0.998i)\)

Particular Values

\(L(\frac{1}{2})\) \(\approx\) \(1.191382446 - 1.135652828i\)
\(L(\frac12)\) \(\approx\) \(1.191382446 - 1.135652828i\)
\(L(1)\) \(\approx\) \(1.121749608 - 0.2058003495i\)
\(L(1)\) \(\approx\) \(1.121749608 - 0.2058003495i\)

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad2 \( 1 \)
3 \( 1 \)
11 \( 1 \)
good5 \( 1 + (0.964 + 0.263i)T \)
7 \( 1 + (-0.761 - 0.647i)T \)
13 \( 1 + (0.0665 - 0.997i)T \)
17 \( 1 + (0.736 + 0.676i)T \)
19 \( 1 + (0.198 - 0.980i)T \)
23 \( 1 + (-0.235 + 0.971i)T \)
29 \( 1 + (-0.00951 - 0.999i)T \)
31 \( 1 + (-0.345 + 0.938i)T \)
37 \( 1 + (-0.466 - 0.884i)T \)
41 \( 1 + (0.290 - 0.956i)T \)
43 \( 1 + (0.0475 - 0.998i)T \)
47 \( 1 + (-0.820 + 0.572i)T \)
53 \( 1 + (-0.0285 - 0.999i)T \)
59 \( 1 + (0.683 - 0.730i)T \)
61 \( 1 + (-0.797 - 0.603i)T \)
67 \( 1 + (0.327 + 0.945i)T \)
71 \( 1 + (-0.993 + 0.113i)T \)
73 \( 1 + (-0.610 + 0.791i)T \)
79 \( 1 + (0.161 - 0.986i)T \)
83 \( 1 + (0.997 + 0.0760i)T \)
89 \( 1 + (0.415 - 0.909i)T \)
97 \( 1 + (0.964 - 0.263i)T \)
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   \(L(s) = \displaystyle\prod_p \ (1 - \alpha_{p}\, p^{-s})^{-1}\)

Imaginary part of the first few zeros on the critical line

−18.43640062629299882313126752495, −18.04824703204854678408365756831, −16.87499462790230165077489006640, −16.443471390157324700935343750715, −16.146765363098929383104126344398, −14.881164518928619648820081873003, −14.49812561165795994329343987733, −13.659314567210829810209262825696, −13.14908159608255321542063932535, −12.24157069764403020095084765124, −11.97624814947682026446101097498, −10.88873202471751142999945809541, −10.03405818821157345302278511105, −9.58315058608205189969269532994, −8.99802240692553333564943345001, −8.26457107247240163641335624406, −7.2837600860281181227287662670, −6.38689326521742612443771540425, −6.04360239077280416043807033935, −5.19850328623849706632028267452, −4.48668914630216927430852480100, −3.40881560693224729829902683059, −2.708846843194575960680347611711, −1.89673316887744785549804151522, −1.10448031816997310692018170328, 0.46098226368933222495511655721, 1.45382385197533836610075521102, 2.354320545747884837384463047582, 3.27379480920299844899090621442, 3.70025477540088559180030576866, 4.92673980540789783345743133171, 5.64726526496893081069415104137, 6.17583916082023106255423301028, 7.06817391195788977640339050100, 7.5635193841672635371858751718, 8.58559284925626701240511699545, 9.36048731725959980771418687242, 10.02143615267248123185979888737, 10.45406709774261144367165545770, 11.14636881766660480705750090183, 12.16701183244097017192586188399, 12.98971313612135161286238186809, 13.279771992632522644756762196868, 14.07606204619836856873271656794, 14.63610248457090757704509581459, 15.61441104015585352306645349476, 16.02726617158875409717387280491, 17.02609003729935179937448032283, 17.53063420313479161844771762390, 17.860153100558071685361305134778

Graph of the $Z$-function along the critical line